looking at the Hyperbolic Exercises- page 90 of the text, q1 tells us we have a H-plane with two points, A and B (in addition to the center-point O) and that all these points rest on the same line.
We are asked to construct the H-line passing through A and B and to explain why this H-line is unique.
Is this simply a line seg thru A, O, and B (thus bisecting the H circle) and so it is unique because it is the only line seg (that first kind of H-line) that can pass through all 3 points?
Will we need to know how to construct Julia Sets on the final exam, or just how to iterate more basic fractals like in the mid-term?
Regarding the question about hyperbolic geometry: you are right. In fact there is only one H-line passing through any two distinct points.
Regarding fractals: You do not need to know Julia sets in the final. A problem involving iterations of fractals (of the type in the midterm) is an option.
In the Fall 2003 Final Exam, we are asked (question 6a) to construct a hyperbolic line perpendicular to a given hyperbolic line (arc). In class we saw how to construct parallel hyperbolic lines, not perpendicular ones. Is a line (type 1 or type 2) perpendicular to the given arc if it touches it?
In class we covered right angles between hyperbolic lines (or lines and circles). That is sufficient. That particular question is very easy, and the construction can be done in less than 5 seconds. The answer to your last question is negative: if a line touches a circle, then the angle between the two is 0.
I am a little confused about the difference between homotopy in 2d versus homotopy in 3d. We covered it a little bit in class but the textbook does not mention it (as far as I can see), I was just wondering if you could clarify the differences between the two a little bit?
In case of homotopy within 2D you are confined within the drawing plane, while homotopy within 3D allows you to get out of that plane and do things in 3 dimensions. For example, a point within a circle can be taken out of that circle (without cutting or gluing) in 3D but not in 2D. (That is, a planar object consisting of a circle and a point is homotopic in 3D to the object in the same plane consisting of a circle and a point out of it. However, these two objects are not homotopic within 2D i.e., within the plane).
Euler Characeristic and 2-manifolds: just want to verify that the Euler Char of a double Torus is actually -2 ?
e(x)= 2-2g(x), and for double torus the genus is 2 (right?) so:
e(x) = 2-2(2) or e(x) = 2-4 = -2?
Anyone? My notes, somehow, look like a crazy person got to them...
referring to the final exam from spring 2003, question 7 a) I am confused as to how we would construct a line parallel to the given h-line?
rephrasing the question posted above, I realize that 7 a) actually refers to a perpendicular line (not parallel). To construct a perpendicular to line L as asked (passing through point A) would one just draw a type one h-line which passes through the center point as well as point A?
No - not so simple in this case. You can find this example solved in the notes in a more general setting (where the two h-lines intersect at any given angle). Here is a solution for this question, in a nutshell: find the center-line of A, and the tangent to the given h-line at A (both constructions were done in class); where they intersect is the center of the circle through A giving the desired h-line.
and this creates an h-line of type 2?....sorry I am still rather confused by this construction, I can not seem to locate the general solving steps in my notes.
Yes. The steps are given above; here it is again.
Step 1. Find the center-line for A (this has been done in both the math and in the art part of the course).
Step 2. Find the tangent at A to the given hyperbolic line (this has also been done in class: join the center of the circle defining the given h-line with A and construct the perpendicular at A).
Step 3: Make a circle centered at the intersection B of the two lines constructed in Steps 1 and 2, and of radius BA. The part of that circle within the hyperbolic plane is the wanted h-line.
Hey, maybe you can answer this quick question because i cant seem to find the answer anywhere. How do you find the tangent of A to get A'?
It is on page 146 in the textbook.